Exponential function

This article is about functions of the form f(x) = abx. For functions of the form f(x,y) = xy, see Exponentiation. For functions of the form f(x) = xr, see Power function.

The natural exponential function y = ex

In mathematics, an exponential function is a function of the form

f(x)=bx{\displaystyle f(x)=b^{x}\,}

in which the input variable x occurs as an exponent. A function of the form f(x)=bx+c{\displaystyle f(x)=b^{x+c}}, where c{\displaystyle c} is a constant, is also considered an exponential function and can be rewritten as f(x)=abx{\displaystyle f(x)=ab^{x}}, with a=bc{\displaystyle a=b^{c}}.

As functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (i.e., its derivative) is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b{\displaystyle b}:

Since changing the base of the exponential function merely results in the appearance of an additional constant factor, it is computationally convenient to reduce the study of exponential functions in mathematical analysis to the study of this particular function, conventionally called the "natural exponential function",[1][2] or simply, "the exponential function" and denoted by

Its ubiquitous occurrence in pure and applied mathematics has led mathematician W. Rudin to opine that the exponential function is "the most important function in mathematics".[3] In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. Such a situation occurs widely in the natural and social sciences; thus, the exponential function also appears in a variety of contexts within physics, chemistry, engineering, mathematical biology, and economics.

Exponential function

Representation

ex

Inverse

ln x

Derivative

ex

Indefinite integral

ex + C

The graph of y=ex{\displaystyle y=e^{x}} is upward-sloping, and increases faster as x{\displaystyle x} increases. The graph always lies above the x{\displaystyle x}-axis but can get arbitrarily close to it for negative x{\displaystyle x}; thus, the x{\displaystyle x}-axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its y{\displaystyle y}-coordinate at that point, as implied by its derivative function (see above). Its inverse function is the natural logarithm, denoted log{\displaystyle \log },[4]ln{\displaystyle \ln },[5] or loge{\displaystyle \log _{e}}; because of this, some old texts[6] refer to the exponential function as the antilogarithm.

Since the radius of convergence of this power series is infinite, this definition is applicable to all complex numbers z{\displaystyle z}. The constant e can then be defined as e=exp⁡(1)=∑k=0∞(1/k!){\textstyle e=\exp(1)=\sum _{k=0}^{\infty }(1/k!)}.

now known as e. Later, in 1697, Johann Bernoulli studied the calculus of the exponential function.[8]

If a principal amount of 1 earns interest at an annual rate of x compounded monthly, then the interest earned each month is x/12 times the current value, so each month the total value is multiplied by (1 + x/12), and the value at the end of the year is (1 + x/12)12. If instead interest is compounded daily, this becomes (1 + x/365)365. Letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function,

The derivative (rate of change) of the exponential function is the exponential function itself. More generally, a function with a rate of change proportional to the function itself (rather than equal to it) is expressible in terms of the exponential function. This function property leads to exponential growth and exponential decay.

The derivative of the exponential function is equal to the value of the function. From any point P on the curve (blue), let a tangent line (red), and a vertical line (green) with height h be drawn, forming a right triangle with a base b on the x-axis. Since the slope of the red tangent line (the derivative) at P is equal to the ratio of the triangle's height to the triangle's base (rise over run), and the derivative is equal to the value of the function, h must be equal to the ratio of h to b. Therefore, the base b must always be 1.

The importance of the exponential function in mathematics and the sciences stems mainly from its definition as the unique function which is equal to its derivative and is equal to 1 when x = 0. That is,

If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth (see Malthusian catastrophe), continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time. Explicitly for any real constant k, a function f: R → R satisfies f′ = kf if and only if f(x) = cekx for some constant c.

Furthermore, for any differentiable function f(x), we find, by the chain rule:

Exponential function on the complex plane. The transition from dark to light colors shows that the magnitude of the exponential function is increasing to the right. The periodic horizontal bands indicate that the exponential function is periodic in the imaginary part of its argument.

As in the real case, the exponential function can be defined on the complex plane in several equivalent forms. The most common definition of the complex exponential function parallels the power series definition for real arguments, where the real variable is replaced by a complex one:

Termwise multiplication of two copies of these power series in the Cauchy sense, permitted by Mertens' theorem, shows that the defining multiplicative property of exponential functions continues to hold for all complex arguments:

In this expansion, the rearrangement of the terms into real and imaginary parts is justified by the absolute convergence of the series. The real and imaginary parts of the above expression in fact correspond to the series expansions of cos⁡t{\displaystyle \cos t} and sin⁡t{\displaystyle \sin t}, respectively.

This correspondence provides motivation for defining cosine and sine for all complex arguments in terms of exp⁡(±iz){\displaystyle \exp(\pm iz)} and the equivalent power series:[10]

The functions exp, cos, and sin so defined have infinite radii of convergence by the ratio test and are therefore entire functions (i.e., holomorphic on C{\displaystyle \mathbb {C} }). The range of the exponential function is C∖{0}{\displaystyle \mathbb {C} \setminus \{0\}}, while the ranges of the complex sine and cosine functions are both C{\displaystyle \mathbb {C} } in its entirety, in accord with Picard's theorem, which asserts that the range of a nonconstant entire function is either all of C{\displaystyle \mathbb {C} }, or C{\displaystyle \mathbb {C} } excluding one lacunary value.

These definitions for the exponential and trigonometric functions lead trivially to Euler's formula:

We could alternatively define the complex exponential function based on this relationship. If z=x+iy{\displaystyle z=x+iy}, where x{\displaystyle x} and y{\displaystyle y} are both real, then we could define its exponential as

where exp, cos, and sin on the right-hand side of the definition sign are to be interpreted as functions of a real variable, previously defined by other means.[11]

For t∈R{\displaystyle t\in \mathbb {R} }, the relationship exp⁡(it)¯=exp⁡(−it){\displaystyle {\overline {\exp(it)}}=\exp(-it)} holds, so that |exp⁡(it)|=1{\displaystyle |\exp(it)|=1} for real t{\displaystyle t} and t↦exp⁡(it){\displaystyle t\mapsto \exp(it)} maps the real line (mod 2π{\displaystyle 2\pi }) to the unit circle. Based on the relationship between exp⁡(it){\displaystyle \exp(it)} and the unit circle, it is easy to see that, restricted to real arguments, the definitions of sine and cosine given above coincide with their more elementary definitions based on geometric notions.

for all complex numbers z and w. This is also a multivalued function, even when z is real. This distinction is problematic, as the multivalued functions log z and zw are easily confused with their single-valued equivalents when substituting a real number for z. The rule about multiplying exponents for the case of positive real numbers must be modified in a multivalued context:

The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. Two special cases might be noted: when the original line is parallel to the real axis, the resulting spiral never closes in on itself; when the original line is parallel to the imaginary axis, the resulting spiral is a circle of some radius.

The power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any Banach algebraB. In this setting, e0 = 1, and ex is invertible with inverse e−x for any x in B. If xy = yx, then ex + y = exey, but this identity can fail for noncommuting x and y.

Some alternative definitions lead to the same function. For instance, ex can be defined as

Given a Lie groupG and its associated Lie algebrag{\displaystyle {\mathfrak {g}}}, the exponential map is a map g{\displaystyle {\mathfrak {g}}}↦ G satisfying similar properties. In fact, since R is the Lie algebra of the Lie group of all positive real numbers under multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie group GL(n,R) of invertible n × n matrices has as Lie algebra M(n,R), the space of all n × n matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.

The identity exp(x + y) = exp(x)exp(y) can fail for Lie algebra elements x and y that do not commute; the Baker–Campbell–Hausdorff formula supplies the necessary correction terms.

Some calculators provide a dedicated exp(x) function designed to provide a higher precision than achievable by using ex directly.[12][13]

Based on a proposal by William Kahan and first implemented in the Hewlett-PackardHP-41C calculator in 1979, some scientific calculators, computer algebra systems and programming languages (for example C99[14]) support a special exponential minus 1 function alternatively named E^X-1, expm1(x),[14]expm(x),[12][13] or exp1m(x) to provide more accurate results for values of x near zero compared to using exp(x)-1 directly.[12][13][14] This function is implemented using a different internal algorithm to avoid an intermediate result near 1, thereby allowing both the argument and the result to be near zero.[12][13] Similar inverse functions named lnp1(x),[12][13]ln1p(x) or log1p(x)[14] exist as well.[nb 1]

^In pure mathematics, the notation log x generally refers to the natural logarithm of x or a logarithm in general if base is immaterial.

^The notation ln x is the ISO standard and is prevalent in the natural sciences and secondary education (US). However, some mathematicians (e.g., Halmos) have criticized this notation and prefer to use log x for the natural logarithm of x.

^Converse; Durrell (1911). Plane and spherical trigonometry. C. E. Merrill Co. p. 12. Inverse Use of a Table of Logarithms; that is, given a logarithm, to find the number corresponding to it, (called its antilogarithm) ...